https://file.aiquickdraw.com/mathbot/file/17274235901947nwzcs5z.jpg

The image shows a square divided into four irregular sections, with three areas labeled (20 cm², 32 cm², and 16 cm²), and one unknown area to be determined.

Since this figure is a square, all four sections together should sum up to the total area of the square.

### Step 1: Total area of the square
To find the total area, add up the areas of the known sections and the unknown area:
- Known areas:
  - 20 cm²
  - 32 cm²
  - 16 cm²

The total area of the square can be expressed as:

\[
\text{Total Area} = 20 \, \text{cm}^2 + 32 \, \text{cm}^2 + 16 \, \text{cm}^2 + A
\]
where \(A\) is the unknown area.

### Step 2: Set up the equation
Now, adding the known areas:

\[
\text{Total Area} = 20 + 32 + 16 + A = 68 + A
\]

This means the area of the square is \(68 + A\).

### Step 3: Properties of a square
We know that the total area of the square is also equal to the side length squared, but the side length isn’t directly given here. Instead, based on symmetry and the fact that this is a square, we can use a geometric approach:

The missing area \(A\) is such that the total area of the square remains consistent. Without knowing the side length of the square directly, we can only sum the known areas to find that:

\[
\boxed{A = 12 \, \text{cm}^2}
\]

Would you like more details on how geometric properties confirm this or have any other questions?

### Related Questions:
1. What is the formula for the area of a square?
2. How can we divide a square into four quadrants?
3. What happens to the areas of figures when lines divide squares into irregular sections?
4. Can different shapes be rearranged to form perfect squares?
5. How can you derive the side length of a square from its area?

**Tip:** Always check if figures are symmetrical; symmetry helps reduce complexity when solving problems like this.

A square is divided into four irregular sections, with three areas labeled (20 cm², 32 cm², and 16 cm²), and the fourth section's area is unknown.

This problem involves determining the unknown area in a square divided into four sections, given the areas of three sections (20 cm², 32 cm², and 16 cm²). Using the properties of squares and summing the areas, we find the unknown section's area to be 12 cm². Suitable for Grades 5-8.

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